Optimal. Leaf size=97 \[ -\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6167, 6141, 653, 192, 191} \[ -\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (a x+1)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 653
Rule 6141
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\left (c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\right )\\ &=-\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {5}{7} \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=-\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=-\frac {2 (1+a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}-\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 96, normalized size = 0.99 \[ -\frac {\sqrt {1-a^2 x^2} \left (-8 a^5 x^5+16 a^4 x^4+4 a^3 x^3-24 a^2 x^2+9 a x+6\right )}{21 a c^3 (1-a x)^{7/2} (a x+1)^{3/2} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 124, normalized size = 1.28 \[ \frac {{\left (8 \, a^{5} x^{5} - 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 9 \, a x - 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (a x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 0.66 \[ \frac {\left (a x +1\right )^{2} \left (8 x^{5} a^{5}-16 x^{4} a^{4}-4 x^{3} a^{3}+24 a^{2} x^{2}-9 a x -6\right )}{21 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 99, normalized size = 1.02 \[ \frac {2}{7 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c\right )}} - \frac {8 \, x}{21 \, \sqrt {-a^{2} c x^{2} + c} c^{3}} - \frac {4 \, x}{21 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} - \frac {x}{7 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 134, normalized size = 1.38 \[ \frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x-1\right )}^3}-\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x-1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}+\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}+\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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